Multiple Knapsack-Constrained Monotone DR-Submodular Maximization on Distributive Lattice --- Continuous Greedy Algorithm on Median Complex ---
Year: 2019
Type: preprint
Abstract: We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Since a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a $1 - 1/e$ approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of a distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions, such that the multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property of the continuous greedy algorithm.
Source: arXiv (Cornell University)
Authors: Maehara, Takanori, Nakashima, So, Yamaguchi, Yutaro
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2959816846